I'm not much of a grammarian, but here's what I think on this. I would use the singular verb since the subject is "one" and "in a million is just an adjectival addition which describes the "one."
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I completely agree with your logic but I am left with a question rattling around my head.

"one in a million" is not a quantity. In the same way neither are 1/4, 0.00001%. So we only know the quantity by virtue of knowing the pool to which the proportion applies.

So: "one in a million people in Britain have naturally green hair" would give a number of around 60 which would not be singular.
Or: "one in a million children have an IQ above 200 so I suspect your child is not some Newtonian reincarnation but simply a child who throws apples at windows" which would give a value of zero.

Thus with "is" being correct only when n=1 and the likelihood being that n is almost anything other than 1 (including zero) I would suggest that it should be "are".

"One in 1 million people is / are affected" and "One person in 1 million is affected" are essentially identical in meaning. The second statement unquestionably requires the singular verb 'is', so it can be argued that the first statement similarly requires it.

However (and notwithstanding Tony's caveat), it seems to me that the nub of the issue is whether the verb should be construed as relating to 'one' or to 'people'.

With a construction like this, where the question of whether or not to pluralize the verb can quite plausibly be argued both ways, I'd say that both the singular form and the plural form would be acceptable to most native speakers, which is ultimately the definitive criterion for what is 'correct' here. I've certainly heard both forms countless times.

I'm sure that I've also used both the singular and the plural construction many times without any qualms or hesitation in either case.

"One in 1 million people is / are affected" and "One person in 1 million is affected" are essentially identical in meaning.

Erik et al, I take your point and I would be interested to hear your thoughts on mine, but other phrases "essentially identical in meaning" include: 0.0001% of people are affected, 1000 in 1 billion are affected and in these cases "are" would seem to be a better answer.

I can see that occasionally "is" is sometimes correct but I can't get it to be usually correct.

The main contender for is being correct is if "One in 1 million people is / are affected" is the whole sentence. If it is only part of a sentence than the argument breaks down. "one in a million people in America are affected" must, surely, require "are".

The illogicality seems to me to come from the idea that "one" can be satisfactorily split from "in one million". I would argue that this is as illogical as taking "one hundred" and splitting the "one" from the "hundred" to make the argument that it is correct to say "one hundred is affected", or similarly with "one sixth".

My contention is that "one in a million" is a factor, a proportion, a multiplier from which to deduce that it has a value of one requires further information and, that the number is unlikely to be one.

The main contender for is being correct is if "One in 1 million people is / are affected" is the whole sentence. If it is only part of a sentence than the argument breaks down. "one in a million people in America are affected" must, surely, require "are".

I don't think that I can agree with you Tony. "one in a million people in America is affected", is perfectly reasonable.

The ambiguity of the singular/plural is demonstrated by sentences such as "honest politicians are one in a million".

Also compare "the chances are one in a million" and "the chance is one in a million".

Applying your strict grammatical logic to similar sentences would also demand "one percent of the schoolchildren is ill", which is strange to the point of absurdity.

From memory (I no longer have any style guides handy), the most reliable style guides allow a certain latitude here. As a general rule, if the thrust of the meaning is "a certain number from a larger number", use the plural. If the focus is on the pure statistical (mathematical) relationship, use the singular.

The whole thing is, of course, affected by the proximity of the plural noun concerned.

"One in a million people in America is affected" does not sound as odd as "one in a million schoolchildren is affected".

"Rules" of language are derived from the way in which language is used, not the other way round. When something is said or written that sounds perfectly reasonable to a wide range of native speakers, but which appears to break a rule, then it is the rule that is wrong or inadequate, not the speakers. The difficulty lies, of course, in determining what a "wide range of native speakers" is...

I was wondering why I have more problem than others with "One in a million people in America is affected". After all other contributors, and my wife - who is the epitome of sagacity on such matters - and I think I have determined the reason!

In discussing it with my wife:
My wife reads it as : One in a million people ... in America ... is affected. Which I can see as being singular.
Whereas I read it as : One in a million ... people in America ... is affected. And the phrase "people in America" is a large group which resolves in my reading to three to four hundred million. So I need to use are, not is.